Problem 53 - Algebra for College Students

Question

Consider the sequence whose \(n\) th term is \(a_{n}=(0.999)^{n}\). a) Calculate \(a_{100}, a_{1000},\) and \(a_{10,000}\). b) What happens to \(a_{n}\) as \(n\) gets larger and larger?

Step-by-Step Solution

Answer

a) \(a_{100} \approx 0.9048\), \(a_{1000} \approx 0.3677\), \(a_{10,000} \approx 0.0045\). b) As \(n\) increases, \(a_n\) approaches 0.

Step 1 - Understanding the Sequence

The sequence is defined by the term formula: \[a_n = (0.999)^n\].This means we're taking 0.999 and raising it to the power of \(n\).

Step 2 - Calculate \(a_{100}\)

Plug in \(n = 100\) into the sequence formula: \[a_{100} = (0.999)^{100}\].Using a calculator or computation tool, find \[a_{100} \approx 0.9048\]

Step 3 - Calculate \(a_{1000}\)

Now, plug in \(n = 1000\): \[a_{1000} = (0.999)^{1000}\].Using a calculator, find \[a_{1000} \approx 0.3677\]

Step 4 - Calculate \(a_{10,000}\)

Finally, plug in \(n = 10,000\): \[a_{10,000} = (0.999)^{10,000}\].Using a calculator, determine \[a_{10,000} \approx 0.0045\]

Step 5 - Analyze the Behavior as \(n\) Increases

Observe that as \(n\) gets larger, the value of \((0.999)^n\) becomes smaller and closer to 0. Thus, as \(n\) increases infinitely, \[(0.999)^n \rightarrow 0\].

Key Concepts

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. In sequences, this often means that each term gets progressively smaller. For the sequence given by \({a_n = (0.999)^n}\), we see exponential decay because the base, \(0.999\), is less than 1. Each increase in \(n\) results in multiplying by 0.999 again, leading to a smaller and smaller term.

• Example: \(a_1 = 0.999^1 = 0.999\)
• Example: \(a_2 = 0.999^2 = 0.998001\)

Over time, this repeated multiplication by a number less than 1 continually reduces the sequence's value.
Exponential decay is commonly seen in natural processes, such as radioactive decay and cooling of objects.

Limit of a Sequence

The limit of a sequence describes the value that the terms of a sequence approach as the index \(n\) grows towards infinity. For our sequence \(({a_n = (0.999)^n})\), we want to find the limit as \(n \rightarrow \infty.\) For this sequence, calculating terms for increasingly large values of \(n\) demonstrates that \({a_n}\) gets closer and closer to 0.
Mathematically, we write it as: \[\lim_{{n \to \infty}} (0.999)^n = 0\] The term \(a_n\) never actually reaches zero, but gets arbitrarily close to it. This property is useful in various fields, including calculus and differential equations, where knowing the limit helps understand long-term behavior.
The limit of a sequence is crucial in understanding stability and predicting future outcomes in real-world phenomena.

Sequence Behavior

Understanding the behavior of a sequence involves examining its progression and identifying any patterns or trends. For our sequence \({a_n = (0.999)^n}\), the behavior can be summed up in a few key points:

• Each term is slightly smaller than the previous term, indicating a decreasing sequence.
• As \(n\) increases, the rate of decrease becomes less noticeable in the short term but significant in the long term.
• The sequence approaches 0 as \(n\) increases infinitely.

By examining specific values, we can see the trend:

• When \(n = 100\), \(a_{100} \approx 0.9048\)
• When \(n = 1000\), \(a_{1000} \approx 0.3677\)
• When \(n = 10,000\), \(a_{10,000} \approx 0.0045\)

This pattern demonstrates how the sequence quickly shifts from values close to 1 to values approaching 0.
Sequence behavior analysis is essential in mathematics for making predictions and understanding the long-term outcomes of iterative processes.